Symmetry breaking operators for dual pairs with one member compact

M. McKee, A. Pasquale, T. Przebinda

Published 2021-07-20Version 1

We consider a dual pair (G, G'), in the sense of Howe, with G compact acting on $L^2(\mathbb{R}^n)$, for an appropriate $n$, via the Weil representation $\omega$. Let $\tilde{\mathrm{G}}$ be the preimage of G in the metaplectic group. Given a genuine irreducible unitary representation $\Pi$ of $\tilde{\mathrm{G}}$, let $\Pi'$ be the corresponding irreducible unitary representation of $\tilde{\mathrm{G}'}$ in the Howe duality. The orthogonal projection onto $L^2(\mathbb{R}^n)_\Pi$, the $\Pi$-isotypic component, is the essentially unique symmetry breaking operator in $\mathrm{Hom}_{\tilde{\mathrm{G}}\tilde{\mathrm{G}'}}(\mathcal{H}_\omega^{\infty}, \mathcal{H}_\Pi^{\infty}\otimes \mathcal{H}_{\Pi'}^{\infty})$. We study this operator by computing its Weyl symbol. Our results allow us to compute the wavefront set of $\Pi'$ by elementary means.